Triangular condition

Tensors in (5.23) and (5.25) are defined in a pseudostandard phase system (see Introduction, Eq. (5)). The symbol abc in (5.23) means that parameters a, b, c obey the so-called triangular condition with integer perimeter (any number from these three is not smaller than the difference and not larger than the sum of the other two). From the operators introduced we can compose the operator... 

In angular momentum theory a very important role is played by the invariants obtained while summing the products of the Wigner (or Clebsch-Gordan) coefficients over all projection parameters. Such quantities are called 7-coefficients or 3ny-coefficients. They are invariant under rotations of the coordinate system. A j-coefficient has 3n parameters (n = 1,2,3,...), that is why the notation 3nj-coefficient is widely used. The value n = 1 leads to the trivial case of the triangular condition abc, defined in Chapter 5 after formula (5.25). For n = 2,3,4,... we have 67 -, 9j-, 12j-,. .. coefficients, respectively. 3nj-coefficients (n > 2) may be also defined as sums of 67-coefficients. There are also algebraic expressions for 3nj-coefficients. Thus, 6j-coefficient may be defined by the formula... 

Here and further on the symbol (abc) means that parameters a, b and c obey the triangular condition with even perimeter. If in the radial integral there is X[ = nff instead of 2 = ntlji, then this indicates that in the corresponding parts of the integral function / and quantum number l must be replaced by g and V, respectively. Coefficient fk has the form... 

Selection rules for Ek-radiation in the case of one-electron configurations consist of the triangle conditions, which the parameters j j2 and k in the Clebsch-Gordan coefficients must satisfy. The triangular condition il l2k) with the extra requirement that h + h + k be even takes care of the selection rule with respect to the parities of the configurations. 

The genealogy of the terms can be followed in the diagrams shown in Fig. 4. On adding the electron, the triangular conditions should be obeyed in... 

This symbol imposes a triangular condition on Q, K, Q. In the case of half-filled shell states all Mq values must be zero. The triangular condition then becomes more stringent because of the additional requirement that the sum Q + K + Q be even. This implies ... 

Let us start the discussion with 0=0. These resonances can directly decay to the j = 0 rotational state which automatically has the same helicity 0 = 0. The coupling is mainly provided by the V q ° R) potential matrix element [see Equation (11.9)]. Because of the triangular condition j — j < A < j + j following from the Wigner 3,7-symbol,... 

The parameters obtained for all of the lanthanides in dilute acid solution are summarized in Table III. Before appraising the significance of the data it should be noted that what amount to selection rules (triangular conditions on 6-/ symbols involved in the calculations of the matrix elements) determine whether or not [MJ- can have positive values. Thus M. can only have non-zero values for A/ < 2 M4 has nonzero values for A/ < 4, etc. Hence we find that M has the largest number of non-zero matrix elements, M4 has a moderate number of non-zero elements, and Mo has relatively few non-zero matrix elements. It is clear from the data in Table III that is the best determined of the three parameters, as it should be. to, in contrast, is poorly determined in most of the cases—indeed one can generalize that to has a probable magnitude of < I X I0 and makes essentially no contribution to > 90% of the calculated oscillator strengths in dilute acid solutions. Its important role in hypersensitive transitions will be mentioned later. 

Similarly, when the initial sets differ in I, the number of terms in the direct sum is generally determined by the triangular conditions. However, when surface harmonics are coupled, the rule, Eq. (43) with... 

Here the radial integral is defined by (26.4). The selection rules for relativistic Mfc-transitions between one-electron configurations directly follow from the non-zero conditions of submatrix element (27.1). They consist of triangular condition 7 72 and symbols (abc) at radial integrals, ensuring the even values of the perimeter of the corresponding triangle. It follows... 

In the last formula the index j runs over all the meaningful values for which the triangular conditions of 67-symbols are satisfied. For better transparency we use the property that the 67-symbol remains invariant under the interchange of columns so that... 

For the given 67-symbol again a special closed formula exists [8]. The triangular condition for the non-zero 67-symbol yields a restriction J[2 = 712, 7,2 1,7,2 2. 

The triangular conditions for the 3y-symbols yield the restrictions M = M, M 1 and M 2, respectively, and closed formulae exist for all these 37-symbols [16]. The restriction to S = S 1 and S = S 2 originates in the triangular relationships in the corresponding 67-symbols involved in the reduced matrix elements of the second-rank irreducible tensor operators... 

A further simplification follows from the triangular conditions which should be fulfilled for the 67-symbols, i.e. 

The triangular condition for the non-zero 3/ -symbols implies M = M, M 1 while only one of the three terms can contribute. The required 3/-symbols are (Appendix 3)... 

From the fact that / = 0 and from the triangular conditions in the 3j symbol of Eq. (250) 0 < 1 <2, sothatA = 2. The selection rules on k with respect to 1 = 2 are k = 1 or 3. The q values for k = 1 and 3 are found in fhe odd crysfal field pofential terms (see Gorller-Walrand and Binnemans, 1996 Gorller-Walrand and Birmemans, 1998 Prather, 1961 Wyboume, 1965). For the four symmetries considered here the even and odd terms are... 

An important consequence of the matrix element theorem concerns the definition of selection rules. An interaction will be forbidden if the corresponding coupling coefficient in the Wigner-Eckart theorem is zero. The conditions that control the zero values of the coupling coefficients are called triangular conditions, since they involve the combination of three irreps. Two kinds of triangular conditions must be taken into account ... 

Selectivity on the subrepresentations subrepresentations that are defined in a splitting field must obey the triangular conditions for the subduced irreps in the corresponding subgroup. 

Several selection rules follow in application of the triangular conditions to the 3-j and 6-j symbols contained in the theoretical equations. These are ... 

B) The structures the wave vectors of which satisfy the triangular condition... 

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